Topologically multiplexed optical data communication

ABSTRACT

Systems and methods for encoding information in the topology of superpositions of helical modes of light, and retrieving information from each of the superposed modes individually or in parallel. These methods can be applied to beams of light that already carry information through other channels, such as amplitude modulation or wavelength dispersive multiplexing, enabling such beams to be multiplexed and subsequently demultiplexed. The systems and methods of the present invention increase the number of data channels carried by a factor of the number of superposed helical modes.

CROSS-REFERENCE OF PRIOR APPLICATION

This application claims priority to U.S. patent application Ser. No.60/608,657 filed on Sep. 10, 2004 and is incorporated herein byreference.

FIELD OF THE INVENTION

The present invention relates generally to methods for transformingconventional beams of light into helical modes and superpositions ofhelical modes.

BACKGROUND OF THE INVENTION

Optical data communication typically involves modulating the amplitudeand wavelength of a beam of laser light, and detecting that modulationdownstream. The present invention is directed to a complementaryapproach to conveying information on a beam of light based on theproperties of helical optical modes.

A helical mode is characterized by the corkscrew-like topology of itswave fronts, which can be described by a real-valued phase function:φ({right arrow over (ρ)})=lθ  (1)

where {right arrow over (ρ)}=(ρ,θ) is the position in a plane transverseto the beam's axis, with θ being the polar angle, and l is an integralwinding number known as the topological charge that describes the pitchof the helix. This phase establishes the beam's topology through thegeneral expression for the magnitude of the electric field in acollimated beam,E _(l)({right arrow over (ρ)})=υ_(l)({right arrow over(ρ)})exp(iφ({right arrow over (ρ)}))exp(iφ_(l)),  (2)

where υ_(l)({right arrow over (ρ)}) is the real-valued amplitude profileand φ_(l) is an arbitrary constant phase. A general superposition ofhelical modes can be written as $\begin{matrix}{{E\left( \overset{\rightarrow}{\rho} \right)} = {\sum\limits_{l = {- \infty}}^{\infty}\quad{{E_{l}\left( \overset{\rightarrow}{\rho} \right)}.}}} & (3)\end{matrix}$

If it is assumed that all the beams in the superposition have the sameamplitude profile, υ({right arrow over (ρ)}) perhaps with differentamplitudes, α_(l), then $\begin{matrix}{{{E\left( \overset{\rightarrow}{\rho} \right)} = {\sum\limits_{l = {- \infty}}^{\infty}{\alpha_{l}{\upsilon\left( \overset{\rightarrow}{\rho} \right)}{\exp\left( {{i\varphi}\left( \overset{\rightarrow}{\rho} \right)} \right)}{\exp\left( {i\phi}_{l} \right)}}}},} & (4)\end{matrix}$

with normalization Σ_(l=−∞) ^(∞)|α_(l)|²=1. For the practicalapplications, only a limited set of the α_(l) will be non-zero.

SUMMARY OF THE INVENTION

The present invention relates in part to methods for transformingconventional beams of light into helical modes and superpositions ofhelical modes. The present invention also involves detecting helicalmodes and methods for parallel data extraction from superpositions ofhelical modes. The ability to encode and decode information carried in abeam's topology leads naturally to methods for topological datacommunication. A slight elaboration on this theme yields methods formultiplexing and demultiplexing beams of light that also carryinformation through other channels, such as amplitude modulations.

These and other objects, advantages and features of the invention,together with the organization and manner of operation thereof, willbecome apparent from the following detailed description when taken inconjunction with the accompanying drawings, wherein like elements havelike numerals throughout the several drawings described below.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a plot showing radial intensity profiles for superpositions ofhelical modes created from a conventional flat-top beam with acomputer-designated phase-only diffractive optical element;

FIG. 2(a) is a representation of a helical beam with topological chargel being converted to a conventional non-helical beam by a DOE encoding atopological charge of −l; and FIG. 2(b) is a representation of a helicalbeam if the DOE does not exactly cancel the input beam's helicity,wherein the resulting beam still has a dark focus and will not bedetected by the photodetector.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The wave fronts of a helical beam may all meet along the optical axis ata topological singularity known as an l-fold screw dislocation.Conventional beams, by contrast, have no such defect. Introducing such adefect therefore transforms a conventional beam into a helical beam.There are numerous ways to accomplish this transformation and one of themost straightforward methods of transforming a conventional beam into ahelical beam is to sculpt the phase of the conventional beam's wavefronts according to Equation (1) discussed previously. This can beaccomplished by passing the beam of light through a piece of transparentmaterial with a helical surface relief, with the resulting local phaseshift being proportional to the local thickness of the material.

Another method for accomplishing this task is to employ a phase-onlyspatial light modulator (SLM), which is designed to shift the phase ofincident light by a programmable amount at each pixel in atwo-dimensional array. SLMs typically are designed to provide a range of2π radians of phase shift. Because a phase shift of 2π is equivalent toa zero phase shift, the helical profile, which covers an arbitrarilylarge range, can be mapped onto the device's dynamic range with themodulo operator: φ({right arrow over (ρ)}) mod 2π. Light operated on byan SLM picks up the phase factor, exp(iφ({right arrow over (ρ)})) thatdistinguishes the helical beam in Equation (2) from a conventional beam.

The phase pattern that implements this mode conversion is an example ofa phase-only hologram. Whereas an SLM allows for dynamicallyreconfigured holograms, some data communications applications also cantake advantage of various optical elements such as microfabricateddiffractive optical elements (DOEs) with fixed phase transferproperties.

The helical phase function, represented in Equation (1), creates ahelical beam coaxial with the incident conventional beam. This modeconversion may not occur with perfect efficiency. The result maytherefore include an undiffracted portion of the original non-helicalbeam. To avoid this result, it may be desirable to deflect thediffracted helical beam. This can be accomplished by adding a phasefunction encoding a deflection by a wave vector {right arrow over (k)},φk({right arrow over (ρ)})={right arrow over (k)}·{right arrow over(ρ)},  (5)to the phase function encoding the mode conversion. The result is adeflected helical beam, with the undiffracted portion propagating in theundeflected direction.

FIG. 1 is a plot showing intensity profiles for superpositions ofhelical modes created from a conventional flattop beam with acomputer-designed phase-only DOE. The bold curve is computed for asuperposition of eight helical modes with topological charges l=11, 21,31, 41, 51, 61, 71, and 81. The thin curve is for a superposition withthe components at l=21 and 71 excluded. Rescaling the azimuthal averagesby the radial coordinate, r, makes clear that the superposed modes haveequal power.

Superpositions of helical modes are created generally as follows. In ageneral superposition, $\begin{matrix}{{E\left( \overset{\rightarrow}{\rho} \right)} = {\sum\limits_{l = {- \infty}}^{\infty}{\alpha_{l}{\upsilon\left( \overset{\rightarrow}{\rho} \right)}{\exp\left( {{\mathbb{i}}\left\lbrack {{l\theta} + {{\overset{\rightarrow}{k}}_{l} \cdot \overset{\rightarrow}{\rho}} + \phi_{l}} \right\rbrack} \right)}}}} & (6)\end{matrix}$

created from a collimated beam with amplitude cross-section ρ({rightarrow over (ρ)}). Even though the individual modes differ from the inputbeam by a pure phase factor, the sum also features amplitudemodulations. These amplitude modulations can be minimized, but might notbe altogether eliminated, by appropriately selecting the relativephases, φ_(l). Iterative and direct search algorithms are also availablefor computing phase-only holograms that can maximize diffraction intosuch superposed modes.

The data plotted with a bold curve in FIG. 1 were computed for asuperposition of eight modes ranging from l=11 to 81, created from asingle flat-top beam of light with a phase-only hologram. This plotshows the beam's intensity averaged over angles, scaled by thecircumference. Removing two modes from the superposition results in aclearly measurable change in the intensities associated with thosemodes, and a far less substantial change in other neighboring modes'intensities.

Using these methods, a single conventional laser beam or other lightsource can be transformed into a superposition of helical modes, eachtraveling in an independently specified direction. Possible exampleconfigurations include multiple modes propagating in the same direction,or beams with the same topological charge traveling in differentdirections.

A helical mode's topology endows it with an important property for datacommunications. Because all angles are present along the beam's axis,all phases are present. Typically, the resulting destructiveinterference causes the beam to be dark along its axis, regardless ofthe amplitude profile υ({right arrow over (ρ)}). The beam's intensity isredistributed into a ring of light of radius R_(l). The radius of thedark core increases with the beam's topological charge l. In the specialcase that the amplitude profile is that of a Laguerre-Gaussian eigenmodeof the Helmholtz equation, R_(l) is proportional to √{square root over(l)}. This is a conventional concept in the art (see, for example, M. J.Padgett and L. Allen. “The Poynting vector in Laguerre-Gaussian modes.”Optics Communications 121, 36-40 (1995)). More generally, for Gaussianbeams, flat-top beams, and other common profiles, it is conventionalthat R_(l) is proportional to l, (see, for example, J. E. Curtis and D.G. Grier. “Structure of optical vortices.” Physical Review Letters 90,133901 (2003)).

A photodetector whose active area has dimensions substantially smallerthan R_(l) for a given value of l will register no light when directlyilluminated by a helical beam. After operation by the detecting hologramit is then a conventional beam and the beam would now activate thephotodetector. FIG. 2(a) is a representation showing how a helical beamwith topological charge l is converted to a conventional non-helicalbeam by a DOE encoding a topological charge of −l. The resulting l=0mode can be focused onto a photodetector and measured. FIG. 2(b) showshow, if the DOE does not exactly cancel the input beam's helicity, theresulting beam still has a dark focus and will not be detected by thephotodetector.

Recalling that diffractive optical elements are capable of changing abeam's topological charge suggests the method for specifically detectinglight in a particular topological mode depicted in FIGS. 2(a) and (b).The beam of light is operated on by a diffractive optical elementencoding a helical mode with topological charge −l. Any component of thebeam carrying topological charge l is thereby converted to a non-helicalbeam. After operation by the detecting hologram it is then aconventional beam and the beam can be focused to a bright spot. Othermodes with l′≠l will be transformed to helical anodes with topologicalcharge l′−l≠0 and will remain dark on axis. Distinguishing differentmodes can be facilitated by focusing the DOE-transformed beam onto anaperture that will block stray light from undesired modes, therebyimproving the detector's selectivity.

Detecting helical modes does not suffer from limited diffractionefficiency to the same extent that creating them does. In particular, ifsome part of the selected helical mode is not operated on by thedetecting DOE, then that part will not be detected. Other modes will notbe spuriously detected, however, so that faithful mode detection canproceed with an imperfect DOE.

The same selectivity is obtained if the detection DOE also deflects thebeam. In this case, the detector is centered on the deflected beam'swave vector {right arrow over (k)}_(l) rather than the original beam'soptical axis. The detector's DOE can select and deflect different modesinto different directions, each of which can be outfitted with aphotodetector. This permits parallel detection of data encoded insuperpositions of helical beams. For example, the eight modes projectedin FIG. 1 can each be read out separately by such a spatially resolvedparallel topological detector. In practice, each of the superposed modesis deflected into all of the possible output directions. Only theselected mode for a particular direction focuses on the associateddetector, however.

The simplest form of topological data communication is to modulate thetopological charge of a beam of light with a time-varying helicaldiffractive optical element and reading out the result with detectorssuch as those described in the previous section. Data can be encoded inthe time-dependent sequence of topological charges in the beam, with thesimplest modulation involving switching between a state with l=0 andanother with l≠0. A more sophisticated approach encodes data in asequence of several values of l, each of which can be read out with aseparate topological charge detector. In a still more sophisticatedapproach, data can be encoded in multiple simultaneous topologicalchannels using a superposition of helical states such as that in FIG. 1.

The detector used to read out the topological charge also may betime-dependent, opening up the ability to hop among topological datachannels. This may be useful in applications akin to frequency hoppingin secure radio communications.

Beams of light that already carry data through other channels, such asamplitude modulation, wavelength modulation or phase modulation, alsocan be transformed into helical beams and superposed with other helicalbeams. Each data stream then is capable of traveling through aparticular topological channel in parallel with others.

In one implementation, a plurality of information carrying beams allilluminate an appropriately designed diffractive optical element, eachat a particular angle. The diffractive optical element deflects all ofthe beams into one or more selected directions, endowing each beam witha specific topological charge. The result is one or more beams carryinga superposition of different helical modes, each carrying informationencoded in other characteristics of the beam. This type of beam isreferred to as a topologically multiplexed beam.

The multiplexed beam can be demultiplexed with a similar DOE thatdissects the superposed beam into the individual constituents, one pertopological channel. These reconstituted beams can be further analyzedwith other techniques. The simplest implementation of this idea woulduse two copies of the same DOE, one to multiplex the beams, and anotherturned backward to demultiplex it.

The system and method of the present invention can be incorporated intoa number of different applications. For example, a beam that alreadycarries multiple data channels can be taken to undergo wavelengthdivision multiplexing, where multiple wavelengths of light can be passedover a single fiber, to impress upon it a helical phase profile, therebymaking the beam amenable to topological multiplexing. Such a systemwould allow for a significantly increased number of topologicalchannels, resulting in a multiplication of the bandwidth of a particularcommunication channel.

Additionally, the present invention could also be used to create anencryption system. This encryption system may further be high high-speedand/or an all-optical encryption system. Furthermore, the superpositionof topological states itself can be used to convey information. Onecould also therefore encode information in the time-dependentsuperposition of topological modes, in addition to any other informationcarried within the input beams themselves. This can be used, forexample, to maintain an encoded checksum for the data carried on themultiple data channels to authenticate the sender of information. Thisprocess can constitute an additional, potentially secure, data channelin its own right. The present invention may provide for securityadvantages and advances in secure communication which deter or preventintrusion, eavesdropping, or unauthorized access, reception or decodingof transmitted information. The present invention may be used at leastin part for quantum communications, quantum cryptography, encoding ofclassical or quantum information, and in conjunction with varioustransmission media including free space communications.

In other embodiments of the invention various higher-order Gaussianbeams can be used as well as Laguerre-Gaussian (LG) modes, higher ordermodes, and/or orbital angular momentum.

In yet another embodiment the present invention may also employ variousoptical elements which include but are not limited to diffractiveoptical elements (DOEs) and which may further include microfabricateddiffractive DOEs with fixed phase transfer properties.

Further embodiments may include multiple transmitters and/or detectors,adaptive optics for such purposes as correction of degradation intransmission medium (e.g. fiber, air, etc) and may adjust for angularmisalignment or lateral misalignment (e.g. of transmitter and/orreceiver or detector).

While several embodiments have been shown and described herein, itshould be understood that changes and modifications can be made to theinvention without departing from the invention in its broader aspects.Various features of the invention are defined in the following claims:

1. A method of creating a topologically multiplexed light beam,comprising the steps of: providing at least one information-carryinglight beam, each illuminating a first optical element; using a firstoptical element to deflect the at least one information-carrying lightbeam into at least one selected direction; and endowing the at least oneinformation-carrying light beam with a specific topological charge,resulting in the at least one resultant light beam carrying asuperposition of different helical modes.
 2. The method of claim 1 wherethe optical element is a diffractive optical element.
 3. The method ofclaim 1 where illuminating the first optical element is done at adesignated angle.
 4. The method of claim 1 wherein deflection of theinformation-carrying light beam comprises adding a phase functionφk({right arrow over (ρ)})={right arrow over (k)}·{right arrow over(ρ)}.
 5. The method of claim 1 further comprising the step of using asecond optical element to dissect the at least one resultant beam into aplurality of constituent beams.
 6. The method of claim 5 wherein thefirst and second optical elements comprise at least two opticalelements, with the second optical element being oriented backwardsrelative the first optical element on an optical axis.
 7. A method ofproviding topological data communication, comprising the steps of:providing a beam of light; modulating a topological charge of the beamof light with an optical element to provide a resultant data beam; andreading out the resultant beam using a detector.
 8. The method of claim7 where the optical element is a time-varying helical diffractiveoptical element.
 9. The method of claim 7 wherein the optical elementintroduces a superposition of topological charges, and wherein data isencoded in the sequence of topological charges, creating a plurality oftopological data channels.
 10. The method of claim 9 where the sequenceof topological charges is time-dependent.
 11. The method of claim 10wherein the time-dependent sequence of topological charges modulatesbetween a state between a first state and a second state.
 12. The methodof claim 11 wherein the time-dependent sequence of topological chargesmodulates among a sequence of a plurality of values of l.
 13. The methodof claim 7 wherein data from the data beam is encoded in multiplesimultaneous topological channels using a superposition of helicalstates.
 14. The method of claim 7 wherein the detector operates in atime dependent manner.
 15. The method of claim 9 further comprising thestep of moving among the plurality of topological data channels usingthe time dependent detector.
 16. The method of claim 7 wherein thedetector comprises a diffractive optical element.
 17. The method ofclaim 7 wherein the detector comprises a spatially resolved paralleltopological detector.
 18. A method for transforming a beam of light intoa superposition of helical modes of light, comprising the steps of:providing a conventional beam of light; introducing a defect into theconventional beam of light, the defect creating a helical beam of lightcoaxial with the conventional beam of light, the conventional beam oflight including an undiffracted portion; and deflecting the helical beamof light away from the undiffracted portion of the conventional beam oflight.
 19. The method of claim 18 wherein the conventional beam of lightis transformed into a helical beam of light by passing the conventionalbeam of light through a transparent material with a helical surfacerelief.
 20. The method of claim 18 wherein the helical beam of light iscreated by using a phase-only spatial light modulator to shift the phaseof incident light by a designated amount at each pixel in atwo-dimensional array.
 21. The method of claim 20 wherein the helicalbeam of light is deflected by adding a phase function encoding adeflection by a wave vector to the phase shift imparted by the spatiallight modulator.